---
type: claim
domain: internet-finance
description: "Larger service systems need proportionally fewer excess servers due to square-root scaling of variance"
confidence: proven
source: "Ward Whitt, What You Should Know About Queueing Models (2019)"
created: 2026-03-11
supports:
- moderate-scale-queueing-systems-benefit-from-simple-threshold-policies-over-sophisticated-algorithms-because-square-root-staffing-captures-most-efficiency-gains
related:
- littles-law-provides-minimum-worker-capacity-floor-for-pipeline-systems-but-requires-buffer-margin-for-variance
reweave_edges:
- littles-law-provides-minimum-worker-capacity-floor-for-pipeline-systems-but-requires-buffer-margin-for-variance|related|2026-04-18
- moderate-scale-queueing-systems-benefit-from-simple-threshold-policies-over-sophisticated-algorithms-because-square-root-staffing-captures-most-efficiency-gains|supports|2026-04-18
sourced_from:
- inbox/archive/internet-finance/2019-00-00-whitt-what-you-should-know-about-queueing-models.md
---

# Multi-server queueing systems exhibit economies of scale because safety margin grows sublinearly with system size

Queueing theory proves that larger service systems are more efficient per unit of capacity. If a system with R servers needs β√R excess servers for quality-of-service, then doubling the base load to 2R requires only β√(2R) ≈ 1.41β√R excess servers, not 2β√R.

The safety margin grows as the square root of system size, not linearly. This creates natural economies of scale: the proportional overhead for handling variance decreases as systems grow. A system with 100 servers needs ~10% overhead (assuming β=1), while a system with 10,000 servers needs only ~1% overhead.

This explains why:
- Large call centers are more efficient than small ones
- Cloud providers achieve better utilization than on-premise infrastructure
- Centralized service systems outperform distributed ones on pure efficiency metrics
- Pipeline architectures benefit from batching and pooling

The implication for Teleo: as processing volume grows, the relative cost of maintaining service quality decreases. Early-stage over-provisioning is proportionally more expensive than it will be at scale.

## Evidence

Ward Whitt presents this as a fundamental result from multi-server queueing analysis. The square-root staffing principle directly implies sublinear scaling of overhead. The Halfin-Whitt regime formalizes this: utilization approaches 1 at rate Θ(1/√n), meaning the gap between capacity and load shrinks proportionally as systems grow.

This is observable in practice across industries: Amazon's fulfillment centers, telecom networks, and financial trading systems all exhibit this scaling behavior.


### Additional Evidence (confirm)
*Source: [[2025-04-25-bournassenko-queueing-theory-cicd-pipelines]] | Added: 2026-03-16*

M/M/c queue analysis demonstrates that the marginal improvement of worker N+1 decreases as N grows, providing mathematical proof that safety margins scale sublinearly. This is a fundamental property of multi-server queues, not just an empirical observation.

---

Relevant Notes:
- domains/internet-finance/_map

Topics:
- core/mechanisms/_map
- foundations/teleological-economics/_map